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All elementary functions from a single binary operator

arxiv.org

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Re: All elementary functions from a single binary operator

#251

Post by ted_dunning »

cryptonector wrote: post #201
DoctorOetker wrote: post #30

EDIT: please change the article link to the most recent version (as of now still v2), it is currently pointing to the v1 version which misses the figures. I'm still reading this, but if this checks out, this is one of the most significant discoveries in years. Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computati…

Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things. So the next thing might be an efficient EML HW implementation.

You should mark sarcasm as subtle as this.

Re: All elementary functions from a single binary operator

#252

Post by SideQuark »

lemonwaterlime wrote: post #246
SideQuark wrote: post #230

This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it. write x#y for 1/(x-y). x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction. it's common problem to show in any (insert various algebraic structu…

I’m fairly certain that the difference between the approaches is that the f(x,y) function you mentioned requires limits to represent certain concepts while the eml approach is essentially a tree or a chain of computations meant to represent a model of a system.

Computing exp or ln is an infinite series, and vastly more compute. Hiding series behind a name doesn’t make them free to compute.

Re: All elementary functions from a single binary operator

#253

Post by SideQuark »

strbean wrote: post #240
SideQuark wrote: post #230

This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it. write x#y for 1/(x-y). x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction. it's common problem to show in any (insert various algebraic structu…

I think this is the novel bit: > This includes constants such as e, pi, and i ; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions .

And those come from the infinite series needed to compute exp and ln. They’re just as much work either way. The exp and ln way are vastly costlier for every op, including simply adding 1 and 2.

Re: All elementary functions from a single binary operator

#254

Post by SideQuark »

TimTheTinker wrote: post #237
SideQuark wrote: post #230

This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it. write x#y for 1/(x-y). x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction. it's common problem to show in any (insert various algebraic structu…

> This too is universal Could that be used to derive trigonometric functions with single distinct expressions?

The exp and ln are infinite series. Exp is roughly the infinite series for cos AND the infinite series for sin. Hiding that every op is an infinite series behind a name doesn’t make things free. It just makes even trivial ops like 1+2 vastly more work.

Re: All elementary functions from a single binary operator

#255

Post by SideQuark »

LolWolf wrote: post #239
SideQuark wrote: post #230

This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it. write x#y for 1/(x-y). x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction. it's common problem to show in any (insert various algebraic structu…

I don't think this can do any of the "standard" constants or what we generally consider to be closed-form expressions, though ! (E.g., no e, pi, exp, log, etc.)

Yes it can, by using the same infinite series that exp and ln use to compute. This one just costs less in money, hardware, energy, and is faster for basically every basic op.

Re: All elementary functions from a single binary operator

#256

Post by SideQuark »

doctorpangloss wrote: post #236
SideQuark wrote: post #234

Earlier quoted context omitted.

It's math. You can check it yourself instead of this (and many other) thoughtless posts.

i'm mocking the LLM-generated scientific article you're replying to, not you. i'm agreeing with you haha

Do you claim all things you don’t understand are LLMs? This is what I mean by these and many of your other comments being extremely poor quality to the point of deliberate ignorance.

The paper above was published in 2012 [1], so that’s quite a feat for an LLM. This takes about zero effort to check.

Put some thought or effort into your claims; they’ll look less silly.

[1] https://orcid.org/0000-0002-0438-633X

Re: All elementary functions from a single binary operator

#257

Post by SideQuark »

ano-ther wrote: post #235
SideQuark wrote: post #230

This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it. write x#y for 1/(x-y). x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction. it's common problem to show in any (insert various algebraic structu…

Good find. It cites a paper from 1935: https://www.pnas.org/doi/10.1073/pnas.21.5.252 Here is a bit more: https://mathoverflow.net/questions/57465/can-we-unify-additi...

Ooh, that 2nd link has a nice construction by Terry Tao giving a clear way to show infinitely many such functions exist for pretty much any set of operations.

Re: All elementary functions from a single binary operator

#258

Post by anthk »

This like Subleq where you can run EForth and EForth itself self-bootstraps:

https://github.com/howerj/muxleq

PD: you don't need gforth to compile:

        cc -O2 -o muxleq muxleq.c
Edit muxleq.fth, add some goodies by editing these values:

             1 constant opt.multi      ( Add in large "pause" primitive )
    1 constant opt.editor     ( Add in Text Editor )
    1 constant opt.info       ( Add info printing function )
    0 constant opt.generate-c ( Generate C code )
    1 constant opt.better-see ( Replace 'see' with better version )
    1 constant opt.control    ( Add in more control structures )
    0 constant opt.allocate   ( Add in "allocate"/"free" )
    1 constant opt.float      ( Add in floating point code )
    0 constant opt.glossary   ( Add in "glossary" word )
    1 constant opt.optimize   ( Enable extra optimization )
    1 constant opt.divmod     ( Use "opDivMod" primitive )
    0 constant opt.self       ( self-interpreter [NOT WORKING] )
Here are my settings.

Then, create a new ".dec" file (subleq program)

   : ./muxleq muxleq.dec  new.dec
Now, to run EForth everytime:

     ./muxleq new.dec
add these further in muxleq.fth code to have them:

     : d. tuck dabs  type ;
     : */ */mod nip ;
The ideal place would be just below the ': dabs ' defition.

Re: All elementary functions from a single binary operator

#259

Post by [deleted] »

SideQuark wrote: post #253
strbean wrote: post #240

Earlier quoted context omitted.

I think this is the novel bit: > This includes constants such as e, pi, and i ; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions .

And those come from the infinite series needed to compute exp and ln. They’re just as much work either way. The exp and ln way are vastly costlier for every op, including simply adding 1 and 2.

[deleted]

Re: All elementary functions from a single binary operator

#260

Post by [deleted] »

SideQuark wrote: post #256
doctorpangloss wrote: post #236

Earlier quoted context omitted.

i'm mocking the LLM-generated scientific article you're replying to, not you. i'm agreeing with you haha

Do you claim all things you don’t understand are LLMs? This is what I mean by these and many of your other comments being extremely poor quality to the point of deliberate ignorance. The paper above was published in 2012 [1], so that’s quite a feat for an LLM. This takes about zero effort to check. Put some thought or effort into your claims; they’ll look less silly. [1] https://orcid.org/0000-0002-0438-633X

[deleted]
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